I think the best websites to search for apartments in USA are apartments.com and rent.com. There are are other sites like forrent.com, mynewplace.com, move.com etc. But they have more or less the same listings. You have to put the zipcode or name of the city and state where you plan to move and specify the type of apartment you want like no. of bedrooms or baths or other amenities or you can just leave it blank. It will list all the apartments complexes in and around that city. You can make a comparison of all the apartments and according to your buget shortlist. Sometimes an area may not have apartment communities, you may then search for private apartments listing in craigslist.org. Sometimes there maybe brokers fee associated with it. you have to check that.
If you have one small kid only, then you have to check with the apartment management whether you can rent a one bedroom apartment. Some apartment communities do not allow renting a one bedroom apartment if the kid is above 2 years. You have to take 2bedroom apartment. Some include utilities like water, heat, electricity etc. in the rent while some do not. So depending on your budget, check before you make any decision. Rent for different lease periods would be different. Rent for 12 month lease would be less than rent for 1 month lease. You can negotiate for the rent or maybe ask for some discount like first month free. It's best if you have a friend or colleague in the same location where you are moving. He can help you in looking for the apartment, maybe even shortlist before you come here.
You can look for the ratings and reviews of apartments on apartmentratings.com. But sometimes one's experience might be entirely different from what others are going through. So it's best to go with an open mind.